Substructure Topology Preserving Simplification ofTetrahedral Meshes

Fabien Vivodtzev1, Georges-Pierre Bonneau2, Stefanie Hahmann2, and Hans Hagen3

1 CEA/CESTA (French Atomic Energy Commission)2 LJK, University of Grenoble and INRIA

3 University of Kaiserslautern

Abstract. Interdisciplinary efforts in modeling and simulating phenomena haveled to complex multi-physics models involving different physical properties andmaterials in the same system. Within a 3d domain, substructures of lower di-mensions appear at the interface between different materials. Correspondingly,an unstructured tetrahedral mesh used for such a simulation includes 2d and 1dsubstructures embedded in the vertices, edges and faces of the mesh.The simplification of such tetrahedral meshes must preserve (1) the geometry andthe topology of the 3d domain, (2) the simulated data and (3) the geometry andtopology of the embedded substructures. Although intensive research has beenconducted on the first two goals, the third objective has received little attention.This paper focuses on the preservation of the topology of 1d and 2d substruc-tures embedded in an unstructured tetrahedral mesh, during edge collapse simpli-fication. We define these substructures as simplicial sub-complexes of the mesh,which is modeled as an extended simplicial complex. We derive a robust algo-rithm, based on combinatorial topology results, in order to determine if an edgecan be collapsed without changing the topology of both the mesh and all embed-ded substructures. Based on this algorithm we have developed a system for sim-plifying scientific datasets defined on irregular tetrahedral meshes with substruc-tures. The implementation of our system is discussed in detail. We demonstratethe power of our system with real world scientific datasets from electromagnetismsimulations.

1 Introduction

In this paper we introduce a system that is able to robustly preserve surfaces and poly-lines defined as substructures in a tetrahedral mesh simplified by repeated edge col-lapses. This problem originates from an application in electromagnetism, as detailed inthe next section. The surfaces (resp. polylines) we are dealing with consist in a subsetof faces (resp. edges) of the tetrahedral mesh. Thus, collapsing an edge of the meshmay result in modification of the surfaces and polylines. Preserving these substructuresduring simplification is a new topic in the literature. There have several papers thathave tackled a more specific issue: the preservation of boundary surfaces in tetrahedralmeshes. The proposed solutions to boundary surface preservation, however, are oftentoo restrictive. Some systems reject any edge collapses in the neighborhood of bound-aries, while others do not allow collapses between boundary and internal vertices.

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0Author manuscript, published in "Topological Methods in Data Analysis and Visualization Springer (Ed.) (2011)"

Fig. 1. Electromagnetism simulation for furtivity studies on a large irregular tetrahedral meshwhere surface and linear substructures are the heart of the simulation as they influence the fieldpropagation.

In contrast our system is less restrictive: edges may be collapsed anywhere in themesh, even in the neighborhood of substructures, and even between a vertex in thesubstructures and a vertex outside the substructures. This leads to higher simplificationrates. Furthermore the simplification is uniformly spread out across the mesh, regardlessof the underlying substructures.

The main contribution of our paper is a robust algorithm to detect if an edge canbe collapsed without modifying the topology of the mesh and of its substructures. Oursystem combines this topological validity test with simple geometric and numeric errormeasures in order to drive the simplification. However the focus of this paper is clearlyon preserving topology – not on preserving the geometry of the mesh or the numeri-cal values attached to it. As a matter of fact, the numerous geometric error measuresproposed in other papers can be straightforwardly extended to take into account sub-structures in a mesh, whereas the preservation of the topology of the substructures isthe real challenge.

In a previous work [14] we introduced a topological test for preserving polylinesin non-manifold triangular meshes. Our current work extends the algorithms of [14]to tetrahedral meshes with 2D and 1D substructures. This extension is not trivial, bothfrom a theoretical point of view and for the implementation. The latter is done very care-fully in our system. Efficient data structures and algorithms are precisely described. Fol-lowing [14], the mesh is modeled as a simplicial complex extended by new simplicesconnecting the substructures to a dummy vertex. The validity of the edge collapse ischecked in the extended complex using results from [4]. Thus there is only one con-sistent test to ensure the preservation of the topology of the mesh and of all embeddedsubstructures. In particular we do not rely on heuristic solutions that would treat themesh and the substructures independently.

The remainder of the paper is structured as follows. Section 1.1 and 1.2 give someadditional motivation for this work. Section 2 reviews related work. Section 3 presentsthe theory and Section 4 describes the implementation in detail. Section 5 presents

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

Fig. 2. (le f t) Detail of a substructure shown at high resolution only inside a volume of interest.The topology of the mesh and the substructures is preserved everywhere. (right) Multiresolutionvisualization of a thin layer of material with the associated electromagnetic field magnitude.

the integration of the topological test in our visualization system, with examples ofapplication on real world datasets.

1.1 Strength of the Topological Criteria

Geometric error metrics in simplification are coupled with a threshold value that onecan adapt to coarsen or refine the data. No matter how good the error computation is,the decision of removal greatly depends on this threshold which is application depend inorder to handle unexpected cases often met in Scientific Visualization. In contrast, topo-logical tools guaranty the integrity of the data after simplification without integratingglobal information or any metric between elements. Combinatorial Topology criteriaare general to any application domain that uses meshes.

1.2 Electromagnetic Simulation

Electromagnetism is a wide field used in many applications such as electromagneticcompatibility, furtivity, or the modeling of new absorbing media. Stealth technologyrelies on the conception and simulation of new absorbing materials in order to decreasethe signals reflected from the target to the radar receiver. This ability to reflect radar sig-nals is characterized through Radar Cross Section (RCS) represented as a single scalar.However, to minimize this value, designers need to fully understand the electromag-netic field behavior on the object surface and in its interior. Numerical simulations ofthis phenomenon often use volumetric finite element methods coupled with a domaindecomposition.

The main challenges when simplifying these data for multiresolution visualizationcome from the complex topology of the embedded structures and the large amountof data. Material boundaries (e.g. interfaces) and various substructures of different di-mensions (possibly point sources and polylines) that exist in specific layers introducea complex topology. The multiple crossing interfaces can be represented as one non-manifold surface embedded in the volume mesh. The union of all polylines can also be

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

seen as a unique linear structure which intersects the non-manifold surface. Figures 1and 3 illustrate such data.

Also, due to the high frequency of the incident electromagnetic field, the mesh needsto be subdivided to the scale of the wave length leading to millions of cells. Because ofthe thin domain decomposition required by the numerical simulation and the design ofspecific objects, current tetrahedral meshes have only a few tetrahedra in the thicknessof a layer (see Figure 7) even though the entire mesh may be composed of a hundred ofmillions of cells.

Fig. 3. (le f t) Polyline emerging from an orthogonal base plan. The mesh is subdivided aroundthis structure. (center) Linear structure with multiple self intersections twisted around a cylindri-cal mass with orthogonal interfaces. (right) Multiple crossing interfaces of thin material layersand polylines of the structure.

Although we introduced our method on electromagnetism simulations, the exactsame topological test can be apply to any domain where the topology of embeddedstructures need to be preserved trough simplification of the tetrahedral mesh (geology,medical imaging ...

2 Related Work

Multiresolution approaches provide techniques to efficiently and accurately explorelarge unstructured polygonal and volume data in Computer Graphics and ScientificVisualization. In order to create a multiresolution representation, many mesh simpli-fication algorithms have been proposed over the last decade ranging from surface tovolume based methods. As we focus on topology criteria, we will only survey thesecriteria used in previous algorithm and dismissed geometry/attribute error metrics andsurface/volume mesh simplification algorithm [10, 2]. In fact any error metrics can becombined to our algorithm without any side effect in order to improve the attribute orgeometric quality of the simplified mesh.

Among the wide range of simplification methods, only a few of them introducespecific topological tests and almost none of them tackle the problem of linear/surfacefeatures in a volume mesh. Early work on topological criteria has been done on sur-

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

face simplification in [13] where the vertices candidate to the removal are classified byanalyzing their local neighborhood.

With notions taken from the computational topology, [8] has proven that the topol-ogy of a triangular mesh can be preserved, through simplification, if the 3 tests intro-duced in the paper are fulfilled. Later, in the PM method [7], these tests are completedwith local heuristics in order to preserve the topology of discontinuity curves defined ona surface. The author proposes 6 rules based on an enumeration of the vertices aroundan edge representing a linear feature.

[9] successfully simplify non-convex tetrahedral meshes by triangulating the nonconvex regions. The authors check for tetrahedron flipping avoiding topological andgeometrical problems only at the boundary. However, the authors point out that sometopological problems can still be undetected with their test during the simplification.All these previous topological criteria are not general enough to preserve the topologyof both 1D and 2D structures in a 3D mesh.

[4] is the groundbreaking paper on topology preservation. It has introduced the linkconditions to robustly check if an edge collapse preserves the topology of a 3-complex.Our paper makes extensive use of these link conditions (reviewed in Section 3).

Intuitively, the link of a simplex u is the subset of simplicies being in contact withall adjacent simplicies of u and the rest of the mesh. In a 3-complex without boundary,[4] has proven that an edge collapse uv preserves the topology of the complex if theintersection of the vertex links is equal to the edge link. This condition is known asthe link condition. For general 3-complexes, this link condition has to be evaluated ina succession of complexes built by iteratively filtering the simplicies having a simplertopological neighborhood.

Extending the multiresolution representation of [3], [1] first integrates the link con-ditions to preserve the topology of both the interior cells and the boundary of the mesh.[12] also uses the link conditions but simplifies them by assuming that the tetrahedralmesh is manifold. The link condition is computed for a 3-complex without boundaryfor the interior tetrahedra and if a boundary tetrahedron is encountered a second linkcondition for the 2-complex at the boundary is added.

Because none of these papers fullfil our needs we introduced a new algorithm whichsuccessfuly simplifies general simplifical 3-complex in preserving both the topology ofthis complex anf the topology of sub-complexes embedded into it. The strength of thisnew algorithm comes from: (i) its locality, (It only uses the direct neighborhood of thesimplicies) (ii) its unicity, (It can handle any complicated topological neighborhoodregardless of its dimension lower than 3.) and (iii) its exactness (It is based on theoricalresults taken from combinatorial topology rather than local heuristics).

3 Topology preserving edge collapse in an extended 3-complex

3.1 Link conditions in a 3-complex

The link conditions introduced in [4] can be used in a tetrahedral mesh to test if anedge e between two vertices u and v can be collapsed without modifying the topologyof the mesh. The mesh is modeled as a simplicial complex K. The link conditions are

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

expressed using basic operators from algebraic topology. We assume the reader is fa-miliar with elementary notions in this field. Precise definitions of the link Lk T , the starSt T and the closure T of a set T of simplices can be found in [11] and illustrated in[14].

If K is a manifold 3-complex, the link conditions reduces to:

Lk u∩Lk v = Lk e (1)

If the 3-complex is non-manifold, the condition (1) has to be tested in three succes-sive smaller complexes of dimensions 3, 2 and 1. This is detailed in Section 3.3.

3.2 Extending the mesh with the substructures

In order to use this link condition to our setting, the embedded structure need to bevisible from the surrounding mesh. To do so, we increase the neighborhood complexityof simplicies defining the sub-structures as follow. The tetrahedral mesh is modeled asa simplicial complex K. The 2D (resp. 1D) substructures are defined as a set L f (resp.Le) of faces (resp. edges) in K. We define an extended complex K by adding to K thecones from a dummy vertex ω to each simplex in the closure of the substructures, asillustrated in Figure 4. The key idea is to consistently encode the topology of the mesh

Fig. 4. Construction of the extended complex. The red simplicies correspond to the substructuresL f and Le. The yellow simplicies represent the elements added to the original complex K (shownin blue) in order to extend K to K.

and its substructures in a single extended complex.

3.3 Link conditions in the extended complex

To preserve the topology of the mesh and its substructures, we apply the link conditionsin the extended complex defined in Section 3.2. Even if the original mesh is manifold,the extended complex is non manifold. Thus, the link condition (1) must be evaluatedrespectively in the following complexes:

Kω = K∪ (ω ·Bd1 K)Gω = Bd1 K∪ (ω ·Bd2 K)Hω = Bd2 K∪ (ω ·Bd3 K)

(2)

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

where Bdi K is the set of all simplicies of order i or less in K. The link of a simplex inKω , Gω and Hω is respectively noted as Lkω

0 , Lkω1 and Lkω

2 . The precise definitionof the order of a simplex is given in [4]. Simplicies of dimension k in a 3-complexhave an order not greater than 3− k. Therefore Kω is a 3-complex, while Gω containsonly edges and Hω only vertices. Note that our algorithm can deal with non-manifoldtetrahedral meshes. However for clarity reasons, the following discussion is restrictedto the case where the tetrahedral mesh is a manifold with boundary.

In this case, Bd1 K contains the boundary faces of K and the faces of L f . It con-tains also the edges and vertices of these faces. Bd2 K contains the edges of Le andthe edges adjacent to exactly one, or else three or more faces of L f (i.e., the boundaryand the non-manifold edges of L f ). It contains also the vertices of these edges. FinallyBd3 K contains the vertices adjacent to one or else three or more edges of Bd2 K (i.e.the boundary and non-manifold vertices of Bd2 K). This includes vertices at the inter-sections created when a polyline in Le crosses a surface in L f .

By experience, the simplicies of high order correspond to regions of interest thatthe user visualizes in detail. Important features in the mesh (detected by cells markedas simplices of high order) are usually correlated with high variations in the data. Thispoint is clearly illustrated in the electromagnetic simulations where the field greatlyvaries around the linear structure or the points of contact in between this element and amass plan.

4 Implementation of the Topology preservation test

4.1 Datastructure of the embedded simplices

The implementation of the link condition requires to iterate through all adjacent sim-plicies around a vertex and an edge. For memory requirement and efficiency we do notstore explicitly all simplicies (faces, edges and vertices) of our substructures. Instead,as both the substructures and the link simplicies are embedded in cells of the originaltetrahedral mesh, all of them are encoded as relative element indices in a tetrahedron.For each tetrahedron 4 bits (resp. 6 bits) are used to indicate which faces (resp. edges)support a 2D (resp 1D) substructure. Both the triangle mesh connectivity of the 2D sub-structures and the linear mesh connectivity of the 1D substructures are retrieved throughthe tetrahedron mesh connectivity without using additional memory. Moreover, duringthe simplification process, such a datastructure eases and optimizes the updates of thesubstructures after an edge collapse. A modification on the tetrahedral mesh will bedirectly passed on to the substructures, without any additionnal treatment.

4.2 Algorithmic implementation

The algorithm 1 describes the evaluation of the link conditions in order to detect if thecollapse of the edge e = uv preserves both the topology of the mesh and the topologyof any embedded structure. The extraction of the link of a simplex is the most repeatedoperation when evaluating the link conditions. It is also time consuming as the star ofthe reference simplex has to be exhaustively visited.

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

Algorithm 1 IsEdgeCollapsePreservingTopology(K, u, v, e)1: # Lines 2 to 21: Compute Lkω

0 u, Lkω1 u and Lkω

2 u2: for all T in St(u) do3: Add to Lkω

0 u the 2-face of T opposite to u, its 3 edges and its 3 vertices4: for all adjacent face fu to u in T do5: if fu is a face of a substructure then6: A dummy tetrahedron is defined as T ω = ω · fu7: Add to Lkω

0 u the face of T ω opposite to u, its 3 edges and its 3 vertices8: Add to Lkω

1 u the edge of fu opposite to u and its 2 vertices9: end if

10: end for11: for all adjacent edge eu to u in T do12: if eu is an edge of a polyline then13: A dummy triangle is defined as tω = ω · eu14: Add to Lkω

0 u and Lkω1 u the edge of tω opposite to u and its 2 vertices

15: Add to Lkω2 u the vertex of eu opposite to u

16: end if17: end for18: if u is an order 3 vertex then19: Add ω to Lkω

2 u20: end if21: end for22: # To compute Lkω

0 v, Lkω1 v and Lkω

2 v apply lines 2 to 21 after changing u in v.23: # Lines 25 to 37: Compute Lkω

0 e and Lkω1 e (Lkω

2 e is alway empty)24: for all Te in St(e) do25: Add to Lkω

0 e the edge of Te opposite to e and its 2 vertices26: for all f of Te adjacent to e do27: if f is a face of a substructure then28: A dummy tetrahedron is defined as T ω = ω · f29: Add to Lkω

1 e the edge of T ω opposite to e and its 2 vertices30: Add to Lkω

1 e the vertex of f opposite to e31: end if32: end for33: if e is an edge of a polyline then34: Add to Lkω

0 e and Lkω1 e the vertex ω

35: end if36: end for37: # Lines 39 to 41: Evaluate the link conditions38: if Lkω

0 u∩Lkω0 v 6= Lkω

0 e then return FALSE

39: if Lkω1 u∩Lkω

1 v 6= Lkω1 e then return FALSE

40: if Lkω2 u∩Lkω

2 v 6= /0 then return FALSE

41: return TRUE

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

Fig. 5. Contribution to the links from the extended simplicies around a reference simplex (u ore shown in green). ω represents the dummy vertex. The substructures are shown in red and thesimplicies to insert into the links in yellow.

The links are directly obtained without explicitly building the extended complex.In order to understand this optimized implementation, one has to associate the dif-

ferent members of equation (2) to the algorithm steps. For example, starting at line 5and illustrated on figure 5 le f t, if fu is a face of a substructure then the dummy tetrahe-dron T ω increases the order of fu to 1. Thus, fu is included in Bd1 K and contributes toLkω

0 but also to Lkω1 because of the member ω ·Bd1 K of Gω (line 8). The simplices

represented in yellow on the figure 5 middle are the contribution of the triangle tω de-fined on line 13 to Lkω

0 and Lkω1 . This extension increases the order of the edge eu to 2

making it visible in Bd2 K. Thus, line 15 of the algorithm implements the contributionof the simplicies present in the member Bd2 K of Hω . A last example, shown on figure5 right, illustrates the contribution of the dummy tetrahedron T ω introduced on line 29.In this case only the opposite edge (yellow) to e (green) is inserted to the link of Lkω

0 eas no face can be present in the link of an edge.

One has to note that due to the strong theoretical background, the topological criteriais insensitive to the complexity of the neighborhood of the collapse (multiple surfacesand polylines).

5 Applications

5.1 Volumetric Simplification

This section presents various applications of the topology preserving visualization sys-tem introduced in the present paper.

The example on figure 6 shows the simplification of a mesh used in electromag-netism simulation. The object is composed of approximatively twenty thin layer ofmaterials encapsulated in each others. The geometry is meshed with about 2 millions oftetrahedra. In order to study the circulation of the electromagnetic field at the boundaryof each materials, all interfaces are explicitely described as embedded surfaces. Ad-ditional linear structures with self intersections are inserted in the model. Figure 6(b)shows the simplified mesh after removing 90% of the vertices with edge collapses. Thisexample illustrates the strength of the algorithm on data composed of a large amount of

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

(a) Original mesh (b) After 90% of simplification

Fig. 6. Volumetric simplification of a model with multiple substructures. A few material layersare shown as opaque triangle mesh and all interfaces as semi-transparent surfaces. The linearfeatures are in red and their self intersections marked by yellow shperes.

substructures of different dimension with multiple self intersections (e.g. increasing thetopological complexity).

Achieving aggressive simplification rates becomes challenging when dealing withlarge volumetric mesh organised in thin material layers. Figure 7 shows a detail of such

(a) Original mesh (b) After 98% of simplification

Fig. 7. Aggressive simplification of a volume mesh composed of thin material layers. The topo-logical criteria preserve both the thickness of the layers and the topology of the substructureswhile allowing collapses of edges on the substructures.

data where the material layers have no more than 5 tetrahedra in their thickness. Lin-ear features are also defined on specific boundaries. Figure 7(b) shows the result after amassive simplification where 98% of the vertices have been removed by edge collapses.In this example the simplification stopped when all candidate edges are rejected due tothe topological criteria. It is important to note that the topological constraints on the 2Dinterfaces insure the preservation of the volume of each layer (at least 1 tetrahedron inthe thickness and usually no more). Another interesting aspect of this example shows

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

that strong constraints on the substructure topology do not disturb or prevent an highsimplification rate of the substructures. The arrow drawn along a linear feature livingon an interface covers about 16 edges on the initial mesh but only 4 edges after simpli-fication. Thus many edges of the interface and of the polyline have been collapsed.

5.2 Multiresolution Visualization

Fig. 8. (le f t) Variable resolution visualization of a volume mesh with multiple linear features.The topology of the substructures is guaranteed to be preserved. (right) Snapshot of the mul-tiresolution visualization tool to explore simulation data with embedded structures on a desktopPC.

In order to provide a visualization system taking advantage of the substructuretopology preserving criteria, the algorithm has been integrated to an existing multireso-lution framework. The MT library [5], freely available and based on the formal approachof [6], provides the tools to build and exploit a multiresolution representation from asequence of valid transformation on a mesh. MT stores a partial order amongst the mod-ifications into a DAG which can then be querried with static or dynamic criteria toproduce various resolutions. Global or local extractors allow the user to extract meshesat variable resolution centered around a volume of interest (VOI). Figures 8 and 1 showa multiresolution mesh at high resolution only inside a VOI interactively moved bya user. The mesh at low resolution reduces the use of the graphic processor allowingthe application to maintain an interactive frame rate even with an original mesh of sev-eral millions of cells. As the substructure preserving topology criteria have been used tobuild this representation, the topology of the substructures is guaranteed to be preservedon the multiresolution visualizations.

6 Conclusion

This paper has introduced new results for simplifying tetrahedral meshes with 2D and1D substructures. As a specific application our system can simplify multi-material tetra-hedral meshes while accurately preserving the topology of the material parts. To the

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0

best of our knowledge, this is the first paper that deals with the simplification of multi-material tetrahedral meshes. Our system can also handle more general 2D and 1D sub-structures. An application in the field of electromagnetism simulation, from which thesystem originates, has been shown. This application and the technology explained (mul-tiresolution) are part of the strategy of the CEA in order to visualize large data computedon a remote super-computer. While focusing on topology preservation, our system canaccommodate any geometric and numeric error measures as well. Once the simplifica-tion is computed, the system uses the MT library to visualize the mesh with higher preci-sion inside a VOI. In future work we plan to optimize the VOI extraction by developingalgorithms specifically designed for handling tetrahedral meshes with substructures aswell as better attribute and geometric attribute metrics to drive the simplification.

References

1. P. Cignoni, D. Costanza, C. Montani, C. Rocchini, and R.Scopigno. Simplification of tetra-hedral volume with accurate error evaluation. In Proceedings IEEE Visualization ’00, pages85–92, 2000.

2. P. Cignoni, L. D. Floriani, P. Lindstrom, V. Pascucci, J. Rossignac, and C. Silva. Multi-resolution modeling, visualization and streaming of volume meshes. In Eurographics 2004,Tutorials 2: Multi-resolution Modeling, Visualization and Streaming of Volume Meshes. IN-RIA and the Eurographics Association, September 2004.

3. P. Cignoni, C. Montani, E. Puppo, and R. Scopigno. Multiresolution representation andvisualization of volume data. IEEE Transactions on Visualization and Computer Graphics,3(4):352–369, oct–dec 1997.

4. T. Dey, H. Edelsbrunner, S. Guha, and D. Nekhayev. Topology preserving edge contraction.Publications de l’Institut de Mathematiques (BEOGRAD), 66(80):23–45, 1999.

5. L. D. Floriani, P. Magillo, and E. Puppo. The MT (Multi-Tesselation) package. DISI, Uni-versity of Genova, Italy, http://www.disi.unige.it/person/MagilloP/MT, Jan. 2000.

6. L. D. Floriani, E. Puppo, and P. Magillo. A formal approach to multiresolution modeling.In W. Straer, R. Klein, and R. Rau, editors, Theory and Practice of Geometric Modeling.SpringerVerlag, 1997., 1997.

7. H. Hoppe. Progressive meshes. Computer Graphics, 30(Annual Conference Series):99–108,1996.

8. H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle. Mesh optimization.Technical Report TR 93-01-01, Dept. of Computer Science and Engineering, University ofWashington, Jan. 1993.

9. M. Kraus and T. Ertl. Simplification of nonconvex tetrahedral meshes, 2000.10. D. Luebke. A developer’s survey of polygonal simplification algorithms. IEEE Computer

Graphics and Applications, 21(3):24–35, 2001.11. J. Munkres. Elements of algebraic topology. Perseus publishing, Cambridge, 1984.12. V. Natarajan and H. Edelsbrunner. Simplification of three-dimensional density maps. IEEE

Transactions on Visualization and Computer Graphics, 10(5):587–597, 2004.13. W. Schroeder, J. Zarge, and W. Lorensen. Decimation of triangle meshes. Computer Graph-

ics, 26(2):65–70, 1992.14. F. Vivodtzev, G.-P. Bonneau, and P. Letexier. Topology preserving simplification of 2d non-

manifold meshes with embedded structures. The Visual Computer, 21(8):679–688, 2005.

inria

-005

4923

4, v

ersi

on 1

- 21

Dec

201

0